The ring resonator is a rather simple passive photonic component, however the uses of it are quite broad.
The basic concept of the ring resonator is that for a certain resonance frequency, those frequencies entering port 1 on the diagram below will be trapped in the ring of the ring resonator and exit out of port 3. Frequencies that are not of the resonance frequency will pass through to port 2.
Ring resonators can be used for Wavelength Division Multiplexing (WDM). WDM allows for the transmission of information allocated to different wavelengths simultaneously without interference. There are other methods for WDM, such as an Asymmetric Mach Zehnder Modulator.
Here I present one scheme that will utilize four ring resonators to perform wavelength division multiplexing. The fifth output will transmit the remaining wavelengths after removing the chosen wavelengths dependent on the resonating frequency (and actually, the radius) of the ring resonators.
Quantum wells are widely used in optoelectronic and photonic components and for a variety of purposes. Two materials that are often used together are InP and InGaAsP. Two different models will be presented here with simulations of these structures. The first is an InP pn-junction with a 10 nm InGaAsP (unintentionally doped) layer between. The second is an InP pn-junction with 10 nm InGaAsP quantum wells positioned in both the positive and negative doped regions.
Quantum Well between pn-junction
The conduction band and valence band energies are depicted below for the biased case:
The conduction current vector lines:
Title Quantum Wells
# Define the mesh
x.m l = -2 Spac=0.1
x.m l = -1 Spac=0.05
x.m l = 1 Spac=0.05
x.m l = 2 Spac =0.1
#TOP TO BOTTOM – Structure Specification
region num=1 bottom thick = 0.5 material = InP NY = 10 acceptor = 1e18
region num=3 bottom thick = 0.01 material = InGaAsP NY = 10 x.comp=0.1393 y.comp = 0.3048
region num=2 bottom thick = 0.5 material = InP NY = 10 donor = 1e18
# Electrode specification
elec num=1 name=anode x.min=-1.0 x.max=1.0 top
elec num=2 name=cathode x.min=-1.0 x.max=1.0 bottom
#Gate Metal Work Function
contact num=2 work=4.77
models region=1 print conmob fldmob srh optr
models region=2 srh optr
#SOLVE AND PLOT
solve init outf=diode_mb1.str master
output con.band val.band e.mobility h.mobility band.param photogen opt.intens recomb u.srh u.aug u.rad flowlines
method newton autonr trap maxtrap=6 climit=1e-6
solve vanode = 2 name=anode
Quantum Well layers inside both p and n doped regions of the pn-junction
#TOP TO BOTTOM – Structure Specification
region num=1 bottom thick = 0.25 material = InP NY = 10 acceptor = 1e18
region num=3 bottom thick = 0.01 material = InGaAsP NY = 10 x.comp=0.1393 y.comp = 0.3048
region num=4 bottom thick = 0.25 material = InP NY = 10 acceptor = 1e18
region num=2 bottom thick = 0.25 material = InP NY = 10 donor = 1e18
region num=6 bottom thick = 0.01 material = InGaAsP NY = 10 x.comp=0.1393 y.comp = 0.3048
region num=2 bottom thick = 0.25 material = InP NY = 10 donor = 1e18
Capacitance relates two fundamental electric concepts: charge and electric potential. The formula that relates the two is Capacitance = charge / electric_potential.
The term equipotential surface refers to how a charge, if moved along a particular path or surface, the work done on the field is equal to zero. If there are many charges along the surface of a conductor (along an equipotential surface), then the potential energy of the charged conductor will be equal to 1/2 multiplied by the electric potential φ and the integral of all charges along this surface.
Ue = ½ φ ∫ dq.
Given a scenario in which both charge and electric potential are related, we may introduce capacitance. The following formula proves important for calculating the energy of a charged conductor:
Ue = ½ φ q = ½ φ2 C = q2 / (2C).
A parallel plate capacitor is a system of metal plates separated by a a dielectric. One plate of the capacitor will be positively charged, while the other is negatively charged. The potential difference and charge on the capacitor places causes a storage of energy between the two plates in an electric field.
Electric potential can be summarized as the work done by an electric force to move a charge from one point to another. The units are in Volts. Electric potential is not dependent on the shape of the path that the work is applied. Being a conservative system, the amount of energy required to move a charge in a full circle, to return it back to where it started will be equal to zero.
The work of an electrostatic field takes the formula
W12 = keqQ(1/r1 – 1/r2),
which is found by integrating the the charge q times the electric field. The work of an electrostatic field also contains both the electric potential and electric potential energy. Electric potential energy, U is equal to the electric potential φ multiplied by the charge q. Electric potential energy is a difference of potentials, while electric potential uses the exact level of electric potential in the given case.
To calculate electric potential energy, it is convenient to assume that the potential energy is zero at a distance of infinity (and surely it should be). In this case, we can write the electric potential energy as equal to the work needed to move a charge from point 1 to infinity.
We’ll consider a quick application related to both the dipole moment and the electric potential. The dipole potential takes the formula in the figure below. Dipole potential decreases faster with distance r than it would for a point charge.
Consider we have both a positive and negative charge, separated by a distance. When applying supperposition of the electric force and electric field generated by the two charges on a target point, it is said that the positive and negative charges create an effect called a dipole moment. Let’s consider a few example of how an electric field will be generated for a point charge in the presence of both a positive and negative charge. Molecules also often have a dipole moment.
Here, the target point is at distance b at the center between the negative and positive charges. Where both charges are of the same magnitude, both the vertical attraction and repulsion components are cancelled, leaving the electric field to be generated in a direction parallel to the axis of the two charges.
Now, we’ll consider a target point along the axis of the two charges. Remember that a positive charge will produce an electric force and electric field that radiates from itself outward, while the force and field is directed inwards towards a negative charge. We can expect then, that the electric field will be different on either side. We can expect that the side of the positive charge will repel and the negative side will attract. This works, because the distance inverse proportionality is squared, making it so that the effect from the other charge will be less. This is a dipole.
Given how a dipole functions, it would be nice to have a different set of formulas and a more refined approach to solving electric field problems with dipoles. The dipole moment p is found using the formula, p=qI with units Couolumb*meter. I is the vector which points from the negative charge to the positive charge. The dipole moment is drawn as one point at the center of the dipole with vector I through it.
In order to treat the two charges as a center of a dipole, there should be a minimum distance between the dipole and the target point. The distance between the dipole and the target should be much larger than the length l of the magnitude of vector I.
Finally, the formula for these electric fields using a dipole moment are
E1 = 2kep/b13
E2 = 2kep/b23
While the electric force describes the exertion of one charge or body to another, we also have to remember that the two objects do not need to be touching physically for this force to be applied. For this reason, we describe the force that is being exerted through empty space (i.e. where the two objects aren’t touching) as an electric field. Any charge or body or thing that exerts an electrical force, generated most importantly by the distance between the objects and the amount of charge present, will generate an electric field.
The electric field generated as a result of two charges is directly proportional to the electric force exerted on a charge, or Coulomb force and inversely proportional to the charge of the particle. In other words, if the Coulomb force is greater, then the electric field will be stronger, but it will also be smaller if the charge it is applied to is smaller. Coulomb force as mentioned previously is inversely proportional to the distance between the charges. The electric field, E then uses the formula E = F/q and the units are Volts per meter.
By combining both Coulomb’s Law and our definition for the electric field, the electric field can be written as
E1 = ke * q1/r2 er
where er again is the unit vector direction from charge q1.
When drawing electric field lines, there are three rules pay attention to:
- The direction is tangent to the field line (in the direction of flow).
- The density of the lines is proportional to the magnitude of the electric field.
- Vector lines emerge from positive charges and sink towards negative charges.
Adding electric fields to produce a resultant electric field is simple, thanks to the property of superposition which applies to electric fields. Below is an example of how a resultant electric field will be calculated geometrically. The direction of each individual field from the charges is determined by the polarity of the charge.
Electric charge is important in determining how a body or particle will behave and interact electromagnetically. It is also key for understanding how electric fields, electric potentials and electromagnetic waves come into existence. It starts with the atom and it’s number of protons and electrons.
Charges are positive or negative. In a neutral atom, the number of protons in a nucleus is equal to the number of electrons. When an atom loses or gains an electron from this state, it becomes a negatively or positively charged ion. When bodies or particles exhibit a net charge, either positive or negative, an electric force arises. Charges can be caused by friction or irradiation. Electrostatic force functions similar to the gravitational force – in fact the formulas look very similar! The difference between the two is most importantly that electrostatic force can be attraction or repulsion, but gravitational force is always attraction. However for small bodies, the electrostatic force is primary and the gravitational force is negligible.
Charles Coloumb conducted experiments around 1785 to understand how electric charges interact. He devised two main relations that would become Coulomb’s Law:
The magnitude of the force between two stationary point charges is
- proportional to the product of the magnitude of the charges and
- inversely proportional to the square of the distance between the two charges.
The following expression describes how one charge will exert a force on another:
The unit vector in the direction of charge 1 to charge 2 is written as e12 and the position of the two numbers indicates the direction of the force, moving from the first numbered position to the second. Reversing the direction of the force will result in a reversed polarity, F12 = -F21.
The coefficient ke will depend on the unit system and is related to the permittivity:
The permittivity of vacuum, ε0 = 8.85*10^(-12) C^2N*m^2.
Coulomb forces obey superposition, meaning that a series of charges may be added linearly without effecting their independent effects on it’s ‘target’ charge. Coulomb’s Law extends to bodies and non-point charges to describe an applied electrostatic force on an object; the same first equation may be used in this scenario.
Rsoft comes with a number of libraries for real materials. To access these materials, we can add them at any time from the Materials button on the side. However, to build a Multilayer structure that can utilize many materials, select “Multilayer” under 3D Structure Type.
Now, select “Materials…” to add desired materials. Move through the RSoft Libraries to chose a material and use the button in the top right (not the X button, silly) to use the material in the project. Now select OK to be brought back to the Startup Window, where we must now design a layered structure using these materials. Note that while building the layers, you can add more materials.
Selecting “Edit Layers…” on the Startup window brings you to the following window. Here, you can define your layers by selecting “New Layer”. Enter the Height and Material of the layer and select “Accept Layer” and repeat the process until the structure is finished. Select OK when done and select OK on the Startup window if all other settings are complete. This is my structure. Note that my structure size adds up to 1. Remember what the size of your layers are.
Now, design the shape of the structure. I’ve made a rectangular waveguide. What is also important to consider is where the beam should enter the structure. By default, the beam is focused across the entire structure. In the case where a particular layer is meant to be a waveguide, this should be reduced in size. By remembering the sizes of the layers however it will not be difficult to aim the beam at a particular section of the waveguide. For my structure, I will aim my beam at the 0.2 GaInAsP layer. The positioning, width, height, angle and more of the launch beam can be edited in the “Launch Parameters” window, accessible through “Launch Fields” on the right side.
Finally, run a simulation with your structure!
There are cases where you may want to simulate a region of air in between two components. A simple way of approaching this task is by creating a region with the same refractive index as air. The segment between the two waveguides (colored in gray) will serve as the “air” region. Right-click on the segment to define properties and under “Index Difference”, chose the value to be 1 minus the background index.
Properties for the segment:
Symbol Table Editor:
Notice that in the “air” region, the pathway monitor detects the efficiency to be zero, though the beam reconvenes in the waveguide, if the gap is short and the waveguide continues at the same angle, but with losses.
Index grating is a common method to alter the frequency characteristics of light. In Rsoft, a graded index component is found under the “Index Taper” tab when right-clicking on a component. By selecting the tab “Tapers…”, one can create a new index taper.
Here, the taper is called “User 1” and defined by an equation step(M*z), with z being the z-coordinate location.
Selecting “Test” on the User Taper Editor will plot the index function of the tapered component:
The index contour is plotted below:
Here, the field pattern:
Light contour plot:
Launch Fields define where light will enter a photonic device in Rsoft CAD. An example that uses multiple launch fields is the beam combiner.
On the sidebar, select “Edit Launch Fields”. To add a new lauch, select New and chose the pathway. A waveguide will be selected by default. Moving the launch to a new location however will place it elsewhere. Input a parameter other than “default” to change the location, and other beam parameters.
Choosing “View Launch” will plot the field amplitude of the launches. For the plot below, the third launch was removed.
Right-clicking on the structure will give the option to chose the “Combine Mode.” Be sure that Merge is selected to allow waveguides to combine.
The Electro-optic effect essentially describes the phenomena that, with an applied voltage, the refractive index of a material can be altered. The electro-optic effect lays the ground for many optical and photonic devices. One such application would be the electro-optic modulator.
If we consider a waveguide or even a lens, such as demonstrated through problems in geometrical optics, we know that the refractive index can alter the direction of propagation of a transmitted beam. A change in refractive index also changes the speed of the wave. The change of light propagation speed in a waveguide acts as phase modulation. The applied voltage is the modulated information and light is the carrier signal.
The electro-optic effect is comprised of both a linear and non-linear component. The full form of the electro-optic effect equation is as follows:
The above formula means that, with an applied voltage E, the resultant change in refractive index is comprised of the linear Pockels Effect rE and a non-linear Kerr Effect PE^2.
The Pockels Effect is dependent on the crystal structure and symmetry of the material, along with the direction of the electric field and light wave.
When stringing multiple parts together, it is important to check a lightwave system for losses. BeamPROP Simulator, part of the Rsoft package will display any losses in a waveguide pathway. Here we have an example of an S-bend simulation. There appears to be losses in a few sections.
Here, the design for the S-bend waveguide has a few locations that are leaking, as indicated by the BeamPROP simulation.
The discontinuities are shown below, which are a possible source of loss:
After fixing these discontinuities, the waveguide can be simulated again using BeamPROP. In fact the losses are not fixed. This loss is called bending loss.
Bending loss is an important topic for wavguides and becomes critical in Photonic Integrated Circuits (PIC).
Rsoft has the ability to create multilayered devices, as was done previously using ATLAS/TCAD. Rather than defining a structures through scripts as is done with ATLAS, information about the layers can be defined in tables that are accessed in Rsoft CAD.
To begin adding layers to a device, such as a waveguide, first draw the device in Rsoft CAD. To design a structure with a substrate and rib waveguide, select Rib/Ridge 3D Structure Type in the Startup Window.
Next, design the structure in Rsoft CAD.
The Symbol Table Editor is needed now not only to define the size of the waveguide, but also the layer properties. The materials for this waveguide will be defined simply using basic locally defined layers with a user-defined refractive index. Later, we will discuss importing layer libraries to use real materials.To get used to the parameters typically needed for this exercise, layer properties may not need to be defined now before entering the Layer Table Editor.
The Layer Table Editor is found on the Rsoft CAD sidebar. First, assign the substrate layer index and select new later. The layer name, index and height are defined for this exercise.
After layers have been chosen, the mode profile can be simulated.
An interesting feature of BeamPROP simulations and other simulators in the Rsoft packages is that the simulation results can be displayed in a running animation. The following simulation is the result of a simulation of an optical fiber. BeamPROP simulates the transverse field in an animation as a function of the z parameter, which is the length of the optical fiber.
To design an optical fiber component with Rsoft CAD, select under 3D structure type, “Fiber” when making a new project.
To build a cylinder that will be the optical fiber, select the cylinder CAD tool (shown below) and use the tool to draw in the axis that the base of the cylinder is found.
Dimensions of the fiber can be specified using the symbol tool discussed previously and by right-clicking the object to assign these values. Note that animations of mode patterns through long waveguides is not only available for cylindrical fibers. Fibers may consist of a variety of shapes. Multiple pathways may be included. Simulations can indicate if a waveguide has potential leaks in it or the interaction of light with a new surface.